Astérisque
JÜRGEN NEUKIRCH The absolute Galois group of a p-adic number field Astérisque, tome 94 (1982), p. 153-164
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ THE ABSOLUTE GALOIS GROUP OF A p-ADIC NUMBER FIELD
by
Jürgen NEUKIRCH
This is a report on the work of U. Jannsen and K. Wingberg on the explicit determination of the absolute galois group G^ of a p-adic number field k ([5] , [6], [10] ). This description depends upon four invariants q, n, p , a of k which are defined as follows. Let k and k be the algebraic closure and the maximal tamely ramified ex tension of k respectively. As is well known the galois group
9 = G(ktrlk) -1 q is generated by two elements a , T satisfying the relation a T a = T . We put n = [k : Q ] P q = cardinality of the residue class field of k , pS = j/ji _(j.s , (j.s being the group of p-power roots of unity in kt r . a : 153 J. NEUKIRCH THEOREM. - The absolute galois group = G(k|k) is isomorphic to the pro- finite group of n+ 3 generators a » T , x , ... ,x and the following defining re lations A. - The normal subgroup generated by x^, ... , is a pro-p-group. B. - CJTCT * = rq (the "tame relation") C. - There is only one additional relation, namely CTXoa~1=(VT)8xl CV^V*^- Cxn-l'Xn] if n is even, and 1 s axQa — =(XQ,T1 ) gxt p[x1,y ] .[x2,x3] ... [xn_1 , xn] if n is odd. Here we have put , , , h1"1 hp"2 h (x,T)=(o x o TXo T...Xo T') , where TT is the element in Z with nl = Zp . Remarks. - The condition A can easily be replaced by a collection of relations and expresses together with B the selfunder-standing relations in G^ . For the exact definition of the element y occurring in the case of odd n we refer to the original paper [6] . It is of type x^a ' T^ . If for example k|k is re placed by the maximal extension of k(|a ) of odd ramification, then we can take y = V The proof of the theorem is based on the following method. For each finite normal subextension K|k of k^l k the galois group of the maximal p-extension K(p) JK has the structure of a Demus'kin group, given by class field theory. Moreover, a detailed study of the action of the group G = G(K|k) on the group of units of K gives further information on the Demuskin structure under the G-action. These known properties of G^ are now taken as axioms for a new abstract concept, the concept of a "Demuskin formation", which goes already back to Koch [9], and which I therefore would like to call a Koch group over Q . Each such Koch group is endowed with invariants q, n, p , a . In a first step it is proved that two Koch groups with the same invariants are isomorphic. Hereafter Jannsen and Wingberg show, that the abstract pro-finite 154 THE ABSOLUTE GALOIS GROUP OF A p-ADIC NUMBER FIELD group, defined by the generators and relations given in the theorem, is a Koch group with the same invariants as the Koch group and is thus isomorphic to C^. We now explain this procedure in more detail by looking at the following diagram of fields and galois groups. x G K pro-p k K ktr k x K(p) Here K is an open normal subgroup of (J = G(ktr(k) contained in the kernel of a , so that \d is contained in the fixed field K of K . G =Q/K = G(KJk), X^ = G(k|K) an II. - Viewing H*(3C,IFp) as a 1-dimensional subspace of the symplectic space H^fX-J-v , IFp ) we have an isomorphism of G-modules HV.IF ^/H^X.lF )ss IFP[G]N. With respect to the induced non-degenerate bilinear form this G-module is hyper bolic, i. e. , direct sum of two totally isotropic G-submodules. III. - (X*b) 2! M as a G-module. V K 'tor ^ s P Explanation. - Condition I expresses the well known fact that X = Gal(K(p)|K) is a Demuskin group. By class field theory H^X^.IF ) is dual to K*/K*P= (n) /(TT*)* U1/(U1)* where rr is a prime element of k and the group of principal units of K . In this interpretation the cup product goes over into the Hilbert symbol on K*/K*P and U1/(U1)P contains H^K, IFp)"L/H1(3C, IFp) as a subspace of co- 155 J. NEUKIRCH dimension 1 which is isomorphic to IF [G] . The last isomorphism is a result of Iwasawa. The assertion on the hyperbolic property is due to Koch. Taking the conditions I, II, III as axioms we come to the following abstract de finition. Let Q be any profinite group generated by two elements a , T such that ax a and let a : Q -» (Z/ps)* be a character. DEFINITION. - A Koch group over (J (D emu skin formation in Jannsen's and Wingberg's terminology) of degree n , of torsion ps and character a » is a pro-finite group X together with a surjective homomorphism $ : X -> Q such that for each open normal subgroup K of Q in the kernel of a , the condi tions I, II, III hold for X. = $ *(K) , where in III (j. is replaced by the twisted Q -module Z/p (a). For these Koch groups we have now the following uniqueness theorem, which was announced by Koch [9] and was proved in full detail and even larger generality by Wingberg [10] . THEOREM I (Koch, Wingberg) . - Two Koch groups X and Y over (J with the same invariants n , ps and a are isomorphic. We indicate the concrete ideas of the proof, by sticking to the field theoretic situa tion and taking for X the Koch group . The problem is roughly speaking, to reconstruct G^ purely by means of the axioms I, II, III. We look again at the diagram x X k G z -k ktr K. 1 3qa K(p) . The inserted field Ki is explained in a moment. Since Gal(k|ktr) is a pro-p- group it is clear that k = (J K(p) and hence KSk tr= lim Gal (K(p)| k). 156 THE ABSOLUTE GALOIS GROUP OF A p-ADIC NUMBER FIELD This reduces us to the question, in which way the group Gal(K(p) | k) is deter mined by the axioms. To attack this problem we look at the group extension 1 —• Gal(K(p) |K) —• Gal(K(p) |k) —• G—> 1 and we filter the Demuskin group = Gal(K(p) JK) by its central series X^ = X_ , X_ = [ XL~ , X ] . (X^, )P . The field K. in the above diagram is the fixed field of X* , i.e. K.| K. is i JC i l -1 the maximal abelian extension of exponent p s . We now obtain the group extensions 1—• Gal(K.|K) > Gal(K.|k)—> G —• 1 . Since Gal(K(p)|k) = lim Gal(Kjk) we are reduced to the question, how to obtain the group Gal(K.|k) by using only the axioms. This is achieved in successive steps i= 1, 2, ... . To mention one surprinsing fact in advance : It suffices to look only at the cases i=l,2 . Once for these cases the group Gal(K.|k) is deter mined by the axioms it is automatically determined for all i . In the case i=l we have to characterize the group G(Kj|K) as a G-module by the axiorrts and to determine the cocycle of the group extension in H2(G, G(KJK)) . Now G(KjK) is dual to H1(X_.,^/pS) and we have seen in the explanation following the axioms I, II, III that this group is very close to the G-module H1 (K , Z./p8)±/H1(3C , ^/pS) ^ Z/pS[G] . With few additional inves tigations this gives the G-structure of G(K1 |K) . For the further developments however this is not enough. For example, one has to determine H ( X__ , JZ/p ) not only as a G-module but moreover as a symplectic G-module by means of axiom II. Furthermore one has to keep track of that part of G(K^|K) which comes from the torsion part of the abelian made group ~ Gal(K(p)| K)ab . This is achieved by means of the so-called Bockstein operator H1 ( X^ , Z/pS) H2(X^, z/Ps) , the image of which is dual to this torsion part in G(K^|K). Having determined the G-module G(K^|K) in sufficient detail by the axioms we have then to determine the cocycle in H^(G, G(KjjK)) associated to the group extension 1—• GfKjlK) > GCKjJk) > G—• 1 in order to describe the group G(K1 |k) . This is done by going over to a p-Sylow 157 J. NEUKIRCH group G of G , which is cyclic, so that P ? 7 8 8 rT(Gp, G(K1|K))= rT(Gp,K*/K*P )= H°(Gp> K*/K*P ). The cocycle is then represented by a prime element TT of k . The selection of this prime element can be group theoretically interpreted by the selection of a section \ : (Ja b G^a b . In this way G(K^z I| k) is completely characterized by the axioms. Much more complicated is the case i= 2 , i. e. , the study of the group exten sion 1—• G(K2|K)—• G(K2|k) » G-» 1 and we do not go any further into this, since already the case i = 1 has given some indication of the type of necessary investigations. The cases i = 1, 2 have brought us to the following situation. We have the two Koch groups X= \ ' Y ' Q and the normal sub groups ^=$V)= Gal(k|K) , YJC = Y"1(K) Let xL and Y* be the pre-image of and Y* under the canonical sur- jection JV JV JV JV where > Y^, is the i-th group in the central series of the Demus'kin group *K 9 %c • Then X/X^ = Gal(K.|k) . Since for i= 1, 2 we have determined this group completely in terms of the axioms I, II, III, which are satisfied by X as well as by Y , we obtain an iso morphism X/xj.* y/Y^ for i = l,2 . We want such an isomorphism for all i and we have to show inductively that the surjective homomorphism Y X/X* with kernel Y* can be lifted to a sur- jective homomorphism Y -» X/X^ . This leads us to the so-called "imbedding problem" for the group Y , i. e. to the diagram 158 THE ABSOLUTE GALOIS GROUP OF A p-ADIC NUMBER FIELD (1) Y xx xx i+1 1- • x/x x /xx i 1 . E x 1C A "solution" of this imbedding problem is a surjection Y x which inserts X x into the diagram commutatively. We consider also the imbedding problem (2) Y xi+2 1 X/X i+2 X/xJo 1 . 4 x x .i+2 In turns out that the group 4 - Gal(K.+2 K.) is abelian for i >1 and we have the following LEMMA. - If i>l then the imbedding problem (2) has a solution iff the imbed ding problem (1) has a solution. If this lemma is shown, we have an isomorphism x x+ x X/XX for all i , and the theorem is proved. Namely (1) has a solution for i = l , by what has been shown before. Therefore (2) has a solution for i = 1 , and hence (1) has a solution for i = 2 etc. For the proof of the lemma we have to consider the diagram 4+2 Inf xv H2(X/4 X /x. H2(Y , XjJ. x x Inf H2(x/xj, i+1 H2(Y , xl sd 4 x x It is very well known and easy to show that the imbedding problem (1) or (2) has a solution, if the cohomology class associated to the group extension is mapped to zero under Inf . Therefore the lemma would follow immediately if H2(Y __i+2 H2(Y, .i+i 4 4* "5C were an isomorphism. A closer examination shows, that we can replace here Y by the group Y which is the maximal pro-p-factor group of the pre-image Y under Y Y/Y^, of the p-Sylow group of Y/Yj£ • This group Y^ is a 159 /. NEUKIRCH Demuskin group and because of the Poincaré duality the requested bijectivity runs up to the bijectivity of 4 H°(Y , Hom(^ i+2 H°(Y Horn 4+1 xxx P xv xv P "X 5C P which can be directly checked because of the known structure of X^ /X^~2 and the -action on it. This finally proves the theorem. Wingberg's actual proof of the uniqueness theorem is more abstract, but it is perfectly modelled after the field theoretical considerations which I have indicated above. The next step is now to construct an abstract Koch group X with the same invariants n , pS , a as . This is done in the following way. Let be the free pro-finite group of n+1 generators , ... , and let Fn+ 1.. # ^G be the free pro-finite product of Fn+ 1, with G = G(ktr 1Ik ) . We then have an exact sequence 1 -+ Z Fn+1 # Q -> Q -» 1 , where Z is the normal subgroup generated by Zq , . .. , . Let P be the maxi mal pro-p-factor group of Z . The kernel of Z P is normal in * Q and we obtain a commutative diagram 1 Z F *Q (J 1 1 -> P -> F(n+1,Q) -» Q -» 1 where P is the normal subgroup of F(n+1,Q) generated by the images x0»"->xn of Zq, ...,«n. The group F(n+1 , (J) is in a sense universal among the split group extensions of Q by a pro-p-group. We now consider the element -CT s r = Xo (x,T)g [xrx2] [x3,x4]...[xn_1,xn] in F(n+1, Q) (for simplicity only in the case of even n) . If can be shown that r^P. Denoting by < r> the normal subgroup of F(n+1 , Q) generated by r and setting V = P/ 1 V X Q - 1 . THEOREM II. - X is a Koch group over (J of degree n , torsion p and character a . 160 THE ABSOLUTE GALOIS GROUP OF A p-ADIC NUMBER FIELD Clearly, theorem I and theorem II together yield an isomorphism gk = 1 and hence the structure theorem for , since X is constructed in exactly such a way as to satisfy the relations A , B , C in this theorem. The starting point which finally led to the relation r , was the following basic result of Jannsen. In order to get the structure of one has to study an arbi trary finite normal tamely ramified extension K|k and the action of its galois group G on the galois group of the maximal abelian p-extension of K . Via class field theory this amounts to the determination of the group of principal units of K as a module over the group ring Z^[G] . Now is known to be a coho- mologically trivial G]-module. Making a complete classification of cohomo- logically trivial Z^[G]-modules, Jannsen proved that there always exists an exact sequence 0 • 2Z [G] Z [G]n+1 > U1 > 1 , P P so that U has only one defining relation as a Z^[G] module, the image of 1 under p . Translating this back into the language of galois groups this made clear, that there should be only one essential defining relation for the group G^ . It was then the task to find this relation in such a way, that the axioms I, II, III of a Koch group were satisfied. This try enforced the specific shape of the rela tion r , and we give now some indications about how the special nature of r implies the properties I, II, III. s The relation /r has a leading term XQC (XQ , T)^ X^ and a commutator term [x^,x^] ... [x^ ^ , x^] . The leading term is responsible for all assertions not in volving the cupproduct and the commutator term for axiom I, which conerns the Demuskin structure. We consider again the diagram 1 P F(n+1, Q) -» Q 1 1 V • X • Q -> 1 . The abelian made group Va^ = Pa^/ 161 J. NEUKIRCH KE -a p-1 hp-2 h s -0 x1s r = Xo (x T x' T ... xo = X oxn mod [P, P] , o 1 o where X is a certain element in 7L [[Q ]] , and thus Va b ^ n (X^)to / bn/ [[[c]]] + n 7L [[Q ]] , ql i=0 Taking everything mod pS one finds that X has the type of an idempotent 00 -i i . s - D h T , showing that (mod p ) Q acts on x as multiplication by g and T i=0 ° as multiplication by h . This gives the (J-isomorphism , ab V ® Z/PS 7L [[Qz/Ps[[Q] ]] , n ] x . pà i1 Taking now an open subgroup K £ker(a) of Q one proves the exactness of the sequence X,Qp/Z) i H (xx,Qp/Zp + ms H1 p H2^ ,^/p1 7L) 0 and taking Pontrjagin duals this yields the commutative exact diagram 0 -> H2(X .Z/p1^)* xx ab x ab 0 v/[v, Xx] v/cv.x^.] p 2 ~ for every i < s . This proves dim H (X^ , IF ) = 1 and (X^)tor ~ M s • l b ^ ^S The space H (X^ , IF^) is dual to X^ ®Z/p which as a G-module is gene rated bv (7 , x , x„ , ... , x by the above consideration. If v , Y , ... , Y is 7 o 1 n 0" o the dual IF [ G] -basis of H^X-^IF ), then this space has the IF -basis PL J 3C p' * p X^,X0> P » P € G > i = 1» •••»n • This shows dim H (X^ , IF ) = n. / G+2 . The assertions concerning the cupproduct rely on the following general LEMMA. - Let D be a pro-p-group generated by Y^>'Y^ > such that H2(D, IF )- IF . Let D° = D , D1+1 = [D1, D] .(D1)1* be the central series and P P assume that there holds a relation a.p a.. [yi, yi] = 1 mod D yi i i 162 THE ABSOLUTE GALOIS GROUP OF A p-ADIC NUMBER FIELD If X1..., Xm is the dual basis of H1(D, FP) associated to y1,...,ym then XiUxj=aij$, i h1 (x* ' V* "1(SSc • V"* "2(5Sc' V from which one draws all the required properties concerning the cupproduct. This concludes the proof of theorem II. LITERATURE [1] K. IWASAWA, On Galois groups of local fields, Trans. Am. Math. Soc. 80 (1955), 448-469. [2] A. V. JAKOVLEV, The Galois group of the algebraic closure of a local field, Math. USSR-Izv. 2 (1968), 1231-1269. [3] A. V. JAKOVLEV, Remarks on my paper "The Galois groups of the alge braic closure of a local field", Math. USSR-Izv. 12 (1978), 205-206. [4] A. V. JAKOLEV, Structure of the multiplicative group of a simply ramified extension of a local field of odd degree, Math. USSR Sbornik 35 (1979), 581-591. [5] U. JANNSEN, Über Galoisgruppen lokaler Körper, to appear. [6] U. JANNSEN,and K. WINGBERG, Die Struktur der absoluten Galois gruppen p-adischer Zahlkörper, to appear. [7] H. KOCH, Über Galoissche Gruppen von p-adischen Zahlkörpern, Math. Nachr. 29 (1965), 77-111. [8] H. KOCH, Galoissche Theorie der p-Erweiterungen, Berlin 1970. [9] H. KOCH, The Galois group of a p-closed extension of a local field, Soviet Math. Dokl. 19 (1978), 10-13. 163 J. NEUKIRCH [10] K. WINGBERG, Der Eindeutigkeitssatz für Demuskinformationen. To appear. Jürgen NEU KIRCH Mathematisches Institut der Universität 8400 REGENSBURG (Allemagne de l'Ouest) 164